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Title Measuring chronic and transient components of poverty: a Bayesian approach

Author(s) Hasegawa, Hikaru; Ueda, Kazuhiro

Citation Empirical Economics, 33(3): 469-490

Issue Date 2007-11

Doc URL http://hdl.handle.net/2115/48680

Rights The original publication is available at www.springerlink.com

Type article (author version)

File Information Hasegawa_Ueda_EmpEcon.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

https://eprints.lib.hokudai.ac.jp/dspace/about.en.jsp

Measuring Chronic and Transient

Components of Poverty:

A Bayesian Approach

Hikaru HasegawaGraduate School of Economics and Business Administration

Hokkaido University

Kazuhiro Ueda∗

Faculty of Economics

Nihon Fukushi University

First version: November, 2004

Second version: November, 2005

Final version: May, 2006

Abstract

After the publication of Ravallion’s (1988) seminal work on chronicand transient poverty, wide attention has been given to the componentsof poverty. We propose a Bayesian mixture model to measure poverty anddivide it into chronic and the transient poverty using the Foster, Greerand Thorbecke (FGT) measure. These two types of poverty are illustratedusing the Panel Study of Income Dynamics (PSID) data.

Keywords: Birth-death process, Foster, Greer and Thorbecke (FGT)measure, Gibbs sampling, Markov chain Monte Carlo (MCMC).

JEL Classification: C11, C15, D63

1 Introduction

Conquering poverty has been one of the most important problems for not onlynational but also international society. Although there is a huge amount ofinternational and domestic aid for societies and people suffering from poverty,the fact that poverty does not vanish poses difficult questions about how to dealwith it. It is likely that policies and measures aimed at eradicating poverty havenot matched with types of poverty. Policy-makers should notice the character-istics of the type of poverty in question and adopt policies that correspond toit. It has been pointed out recently that there are two components of povertyobserved. They are “transient poverty” and “chronic poverty.” In Jalan and

∗Address for correspondence: Faculty of Economics, Nihon Fukushi University, Okuda,Mihama-cho, Chita-gun, Aichi, 470-3295, Japan. (e-mail: [email protected])

1

Ravallion (1998), transient poverty is considered as poverty that arises due tovariability in income or consumption over time and chronic poverty is consideredas poverty that persists in mean consumption over time.

In the seminal study by Ravallion (1988) components of poverty are distin-guished by examining the relationship between risk and poverty. Households inthe agriculture sector, for example, face risk due to weather that causes vari-ability in their income. When an index that measures poverty, the headcountratio for example, rises above, or falls below, the poverty line as a result ofsocial mobility in the population, we cannot say definitely whether the changeis led by households which cross the poverty line temporarily because of the riskthey face. Ravallion (1988) examines, “whether risk induced welfare variabilityincreases or decreases aggregate poverty,”1 and shows the importance of distin-guishing transient poverty from chronic poverty. The article defines these twocomponents of poverty and presents a method for measuring them empiricallywith panel data for households in dry regions of Central India by using thepoverty measure proposed by Foster, Greer ant Thorbecke (FGT) (1984).

Thereafter, some empirical analyses to distinguish transient poverty fromchronic poverty have been presented. Since considerable transient poverty isfound in the consumption panel data of Hungarian households used in Ravallionet al. (1995) and that of the region of rural China in Jalan and Ravallion(1998), it has been confirmed that differences in the two types of poverty play animportant role when choosing a policy for dealing with poverty. Gibson (2001)proposed a new method for decomposing cross-sectional poverty estimates intochronic and transient components using household survey data from Papua NewGuinea.

Most of the earlier studies, however, only report values for three types ofpoverty measures descriptively, and do not use statistical inference. The pur-pose of this article is to introduce a statistical model and propose a Bayesianapproach for estimating chronic poverty and transient poverty. We begin byintroducing a multivariate distribution for panel data on income. Since themultivariate distribution may possess a complicated form, we use a finite mix-ture of normal distributions for income.2 Ferguson (1983) suggests that anarbitrary probability density function on the real line can be approximated bya countable mixture of normal densities. We are, therefore, able to obtaina rich class of distributions for the income distribution. The finite mixturemodel in the Bayesian context was developed by Diebolt and Robert (1994).Diebolt and Robert (1994) assume that the number of components, say k, isknown. Recently, Richardson and Green (1997) and Stephens (2000) considerthe case where k is unknown. Richardson and Green (1997) use the reversiblejump method (Green, 1995) that requires the complex calculation of a Jaco-bian matrix, while Stephens (2000) proposes an alternative to the reversiblejump method by using the birth-death process. In this article, the method inStephens (2000) is used for practical reasons.3

The Bayesian approach has several advantages when estimating poverty. In

1See Ravallion (1988, p.1172).2For a mixture model for income distribution, see Pittau and Zelli (2004) and Pittau (2005).3As Hurn et al. (2003) comment, both methods are equally valid on theoretical grounds.

However, the birth-death procedure avoids the complex calculation of a Jacobian that isrequired in the reversible jump method. Furthermore, the implementation of the birth-deathprocedure is independent of the MCMC algorithm (Hurn et al., 2003, p.70).

2

the context of survey data, observed income data usually suffers from survey er-rors, such as interviewer’s error (Groves, 1989). This corresponds to the ‘errors-in-variables (EIV) model’ in econometrics. There are two ways of incorporatingthe necessary information: the non-Bayesian and Bayesian approaches. In thenon-Bayesian approach, the EIV model can be estimated using instrumentalvariables (IV). In the Bayesian approach, no additional variation data (i.e., IV)are required and a prior information on errors is incorporated instead.4 Thisenables us to measure poverty based on not only observed data, but also thetrue income excluded by errors in the observed data. On the contrary, it isdifficult to calculate poverty measure from unobserved true income using thenon-Bayesian approach.

Our Bayesian approach also has the advantage that the model permits het-erogeneity in individual incomes. This approach divides individual income intoseveral income groups as part of the estimation procedure. In this sense, thepossibility of dependency among individual incomes exists, although they are as-sumed to be independent.5 Bayesian posterior analysis is often associated witha computational burden. However, using a recent development in the Markovchain Monte Carlo (MCMC) method, posterior analysis can be implementedeasily.

We estimate the FGT poverty measures for total, chronic and transientpoverty using samples from the posterior distributions of the observed and un-observed data. In order to verify the effectiveness of our method, we definethe posterior intervals. These intervals include poverty measures by Ravallion’smethod. The chronic poverty measure based on the posterior results for unob-served true data can be regarded as a poverty measure for a stationary state.Comparing this chronic poverty and the total poverty derived from the observeddata, we can measure the difference between total poverty in the data, which ispredicted to be observed, and true poverty.

This article is organized as follows. In Section 2, we present a Bayesianmixture model for density estimation. In Section 3, the poverty measures forthree types of poverty (total poverty, chronic poverty and transient poverty) aredescribed. In Section 4, an application of our approach to the Panel Study ofIncome Dynamics (PSID) data is provided for illustration. Finally, in Section5, a brief summary and some extensions of our approach are given.

2 Bayesian Mixture Model

2.1 The Basic Model and Ravallion’s Decomposition

Let Xit denote the income of the ith individual (or household) at period t. Weuse xit as a logarithm of Xit, that is xit = logXit. It is assumed that xit isgenerated by the following model:

xit = μi + uit, i = 1, · · · , n; t = 1, · · · , T. (1)

In (1), μi is the ith individual’s steady-state logarithm of income and is assumedto be constant over time. uit is a transitory component that determines thedeviation from the steady-state logarithm of income over time.

4See, for example, Florens et al. (1974), and Hasegawa and Kozumi (2001).5See Hasegawa and Kozumi (2003, p.278) for more details on this point.

3

Ravallion (1988) proposed a method for decomposing total poverty intochronic and transient components. Following Ravallion (1988) and Jalan andRavallion (1998), we define total, chronic and transient poverties. Let Υ denotea poverty line and υ = log Υ. We also use the poverty measure proposed byFoster, Greer and Thorbecke (FGT) (1984). The discrete formulation of FGTat period t is defined as

Π(Υ : t) =1n

∑Xit<Υ

(1 − Xit

Υ

)ζ

=1n

∑xit<υ

(1 − exit−υ

)ζ, t = 1, · · · , T,

(2)

where ζ ≥ 0 is a sensitivity parameter of the poverty measure. Since the steady-state logarithm of income μi is unobservable, Ravallion’s decomposition replacesμi with Xi =

∑Tt=1Xit/T . Thus, total, chronic and transient poverties are

defined as follows:

Total poverty: ΠF (Υ) =1T

T∑t=1

Π(Υ : t) (3)

Chronic poverty: ΠC(Υ) =1n

∑Xi<Υ

(1 − Xi

Υ

)ζ

(4)

Transient poverty: ΠT (Υ) = ΠF (Υ) − ΠC(Υ). (5)

The poverty measures defined in (3), (4) and (5) are descriptive. Alterna-tively, in this article we begin by introducing a statistical model for xit. Letdefine xi = [xi1, · · · , xiT ]′. It is difficult to estimate a multivariate density of xi

using a single distribution. Therefore, we assume that the distribution of xi isa mixture of normal distributions with k components. That is,

xi|{wj , μj ,Σj}j=1,··· ,k ∼k∑

j=1

wj N (μj ,Σj),k∑

j=1

wj = 1, i = 1, · · · , n. (6)

Once we have estimated the income distribution of xi, we can use the distri-bution for calculating Ravallion’s poverty indices. In the next subsection, weextend the mixture model (6) to a model with errors-in-variables and presentthe estimation method by using simulation-based Bayesian statistics.

2.2 Bayesian Mixture Model with Errors-in-Variables

Observed income data are collected from surveys, so that the data usually suffersfrom survey errors, such as interviewer’s errors and errors due to respondents.6

We, therefore, introduce explicitly the errors in the observed income. Now, it isconsidered that xit in (6) is logarithm of unobserved true income and observedincome is denoted by Yit. Let us define yit = logYit, and assume that yit isgenerated by the following model:

yit = xit + εit, i = 1, · · · , n; t = 1, · · · , T, (7)

6See, for example, Groves (1989).

4

where εit is an error term.7 Defining yi = [yi1, · · · , yiT ]′ and εi = [εi1, · · · , εiT ]′,(7) is summarized as

yi = xi + εi, i = 1, · · · , n. (8)

Here xi is not a vector of observations, but can be considered as a vector ofunobserved parameters. So in our Bayesian context, (6) is regarded as a priorof xi. We also assume that εi is normally distributed. That is,

xi|θ ∼k∑

j=1

wj N (μj ,Σj),k∑

j=1

wj = 1, i = 1, · · · , n (9)

εit|θ ∼ N (0, σ2t ), i = 1, · · · , n; t = 1, · · · , T, (10)

where θ is a vector of other parameters except for xi’s. To complete the Bayesianmodel, we introduce the following prior distributions:8⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

μj ∼ N (ξ, κ−1)Σ−1

j |β ∼ W(2α, (2β)−1)β ∼ W(2g, (2h)−1)w ∼ D(δ, · · · , δ)σ−2

t ∼ Ga(a, b),

(11)

where w = [w1, · · · , wk]′, W(a,A) denotes a Wishart distribution with degreesof freedom a and scale parameters matrix A, D(δ1, · · · , δk) denotes a Dirichletdistribution and Ga(a, b) denotes a gamma distribution with shape parametera and scale parameter b.

When considering a mixture distribution, it is convenient to introduce latentvariables zi with a probability mass function:

P(zi = j|θ) = wj , i = 1, · · · , n; j = 1, · · · , k.

Conditional on z = [z1, · · · , zn]′, the distributions of x1, · · · , xn are assumed tobe

p(xi|zi = j, θ) = N (xi|μj ,Σj), i = 1, · · · , n.

Integrating out zi, we have the model (9):

p(xi|θ) =k∑

j=1

P(zi = j|θ)p(xi|zi = j, θ) =k∑

j=1

wj N (xi|μj ,Σj).

Using the latent variables z, we can classify x1, · · · , xn. Let us define a setIj = {i|zi = j} and a number of observations allocated to class j, say nj =#{i|zi = j}. We can derive the full conditional distributions (FCD) of θ, xi and

7This specification is familiar in the literature on demand analysis. See, for example,Lewbel (1996) and Hasegawa and Kozumi (2001).

8These prior distributions are same as those in Stephens (2000).

5

zi (i = 1, · · · , n) as follows: for i = 1, · · · , n; t = 1, · · · , T ; j = 1, · · · , k,

P(zi = j|θ, xi, yi) =wj N (xi|μj ,Σj)k∑

l=1

wl N (xi|μl,Σl)

(12)

μj | · · · ∼ N((κ+ njΣ−1

j

)−1 (κξ + njΣ−1

j xj

),(κ+ njΣ−1

j

)−1)

(13)

Σ−1j | · · · ∼ W

⎛⎜⎝2α+ nj ,

⎡⎣2β +

∑i∈Ij

(xi − μj)(xi − μj)′

⎤⎦−1⎞⎟⎠ (14)

β| · · ·W

⎛⎜⎝2g + 2kα,

⎡⎣2h+ 2

k∑j=1

Σ−1j

⎤⎦−1⎞⎟⎠ (15)

w| · · · ∼ D(δ + n1, · · · , δ + nk) (16)

σ−2t | · · · ∼ Ga

(a+

n

2, b+

12

n∑i=1

(yit − xit)2)

(17)

xi| · · · ∼ N((

Σ−1j + Ψ−1

)−1 (Σ−1

j μj + Ψ−1yi

),(Σ−1

j + Ψ−1)−1), (18)

where Ψ = diag{σ21 , · · · , σ2

T }, xj =1nj

∑i∈Ij

xi, and ‘| · · · ’ denotes conditioning

on the values of all other parameters.Since the data y and the predictive data y are conditionally independent

given θ∗ = {θ, x, z}, the predictive distribution can be written as follows:

p(y|y) =∫p(y|θ∗)p(θ∗|y)dθ∗ (19)

Therefore, p(y|θ∗) can be used to estimate poverty indices for y.The degrees of heterogeneity of income are usually unknown. The number

of components in the mixture model k is, therefore, also unknown. When k isunknown, we can use the reversible jump method (Richardson and Green, 1997)and its alternative proposed by Stephens (2000), which is based on the birth-death process. Following Hurn et al. (2003), we use the birth-death processapproach for practical considerations (see the footnote 4). Stephens’ (2000)algorithm will be discussed briefly in the appendix.

3 Poverty Measures

We use the poverty measure proposed by Foster, Greer and Thorbecke (FGT)(1984), as mentioned in Section 2.1. The continuous formulation of FGT isdefined as

Π(υ) =∫ υ

−∞(1 − ey−υ)ζdF (y), (20)

6

where Υ denotes a poverty line and υ = log Υ. The discrete formulation of FGTwith sample size N corresponding to (20) is defined as

Π(υ) =1N

∑yi<υ

(1 − eyi−υ

)ζ =1N

∑Yi<Υ

(1 − Yi

Υ

)ζ

. (21)

Applying the discrete formulation of FGT in (21) to Yit (i = 1, · · · , n; t =1, · · · , T ), we define the sample FGT on Y at period t by

Πy(Υ : t) =1n

∑Yit<Υ

(1 − Yit

Υ

)ζ

, t = 1, · · · , T. (22)

Ravallion’s total, chronic and transient poverties on Y are written as follows:

Total poverty: ΠemF (Υ) =

1T

T∑t=1

Πy(Υ : t) (23)

Chronic poverty: ΠemC (Υ) =

1n

∑Yi<Υ

(1 − Yi

Υ

)ζ

(24)

ΠemT (Υ) = Πem

F (Υ) − ΠemC (Υ), (25)

where Yi =∑T

t=1 Yit/T .Now, we define the total, chronic and transient poverties using the posterior

results. First, we simulate x = [x1, · · · , xT ]′ from its FCD and define the povertymeasure based on xt at period t as follows:

Πx(υ : t) =∫ υ

−∞(1 − ext−υ)ζdF (xt), t = 1, · · · , T.

Using Πx(υ : t), we define total poverty as follows:

Total poverty based on x: ΠpostF (υ) =

1T

T∑t=1

Πx(υ : t). (26)

Furthermore, we calculate ¯x =∑T

t=1 xt/T and define chronic poverty as theFGT of ¯x:

Chronic poverty: ΠpostC (υ) =

∫ υ

−∞(1 − e¯x−υ)ζdF (¯x) (27)

Finally, we define transient poverty in a manner similar to (25):

Transient poverty based on x: ΠpostT (υ) = Πpost

F (υ) − ΠpostC (υ). (28)

The poverty measures based on y are also available from the predictive dis-tribution. That is, we simulate y = [y1, · · · , yT ]′ from p(y|θ∗) and define apoverty measure at period t as follows:

Πy(υ : t) =∫ υ

−∞(1 − eyt−υ)ζdF (yt), t = 1, · · · , T.

7

Using Πy(υ : t), we define total poverty as follows:

Total poverty based on y: ΠpostFy (υ) =

1T

T∑t=1

Πy(υ : t). (29)

We define the transient poverty based on y:

Transient poverty based on y: ΠpostTy (υ) = Πpost

Fy (υ) − ΠpostC (υ). (30)

The discrete formulation of FGT with sample size N defined in (21) converges tothe continuous formulation defined in (20) as N → ∞. Therefore, we calculatethe discrete formulations of (26) to (30) for a large number of N . Furthermore,since the values of (26) to (30) are obtained for each iteration in the MCMCsimulation, statistical inference can be carried out on the three types of pover-ties.

4 Empirical Example

4.1 Model Specification and Data

For an illustration of our Bayesian approach, we use the Panel Study of IncomeDynamics (PSID) for waves 21–26 (1988–1993) of family data.9 There are 5, 371records for the heads of family interviewed successively during the period, andwe select 1, 532 records satisfying the following conditions from them : a) therecord is for a head of family, b) the data for annual family income, familycomposition, and the number of children by sex and age are available, c) thefamily consists of a husband, a wife and a child/children, d) the family incomein a record is greater than 1 and an exact value.

We use “total family money income” as family income, which is sum of thetaxable incomes of the head and wife, total transfers to the head and wife, tax-able prorated income of others, and total prorated transfers of others. Althoughit is common that heads in the records we pick up have a family that consistsof a married-couple and a single child or children, family size can vary with thenumber of children. This raises a problem in how to compare the poverty ofpeople whose family sizes are different, and thus we have to count how manyadults a child or two children correspond to. Therefore, it is required thatthe per capita income be adjusted for the demographic differences to overcomesuch demographic difficulties. In order to calculate per capita income deflatedby equivalence scale from family income, we use the following formulation forequivalence scale proposed by Citro and Michael (1995):

scale value = (A+ PK)F , (31)

where A is the number of adults in a family, K is the number of children, eachof whom is treated as a proportion P of an adult, and F is the scale economyfactor.10 The values we use for both P and F are 0.7, as recommended in Citro

9We use the data for 1988–93 to estimate poverty measures for 1988–92, since we needthe income data for 1988–1992, but the PSID income data for a given year give data for theprevious year.

10See Citro and Michael (1995, p.161).

8

and Michael (1995). The income for estimation is converted into real incomeusing the Consumer Price Index (CPI).11

Tables 1 and 2 show summary statistics and the correlation matrix of thedata, respectively. From Table 2, we can observe that the logarithms of thereal income deflated by equivalence scale for the respective years are highlycorrelated. This fact justifies the formulation of a multivariate distribution ofx in (6) or (9). Figure 1 shows the histograms of y = log Y for each year.12

Table 3 shows the total, the chronic and the transient poverty in the data usingthe method in Ravallion (1988). In this table, we use ζ = 0, 1 and 2, whichcorrespond to the headcount ratio, the poverty gap and the squared povertygap, respectively.

We set the hyperparameter values as follows:13

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ξ =

[miny1 +

R1

2, · · · ,minyT +

RT

2

]′κ = diag

{0.1R2

1

, · · · , 0.1R2

T

}

α = 6.0, g = 0.1α, h = diag{

100gαR2

1

, · · · , 100gαR2

T

}δ = 1.0, a = 2.0, b = 0.002, λ = 1 (birth rate),

(32)

where

maxyt

= max{y1t, · · · , ynt}, minyt

= min{y1t, · · · , ynt}, Rt = maxyt

−minyt

.

The initial value of k is set as k = 1. For a large sample size N , the discreteformulation of FGT converges to the continuous formulation. We can simulatethe samples with a large N from the posterior predictive distributions. We setN = 20, 000. The MCMC simulation was run for 40, 000 iterations and the first10, 000 samples were discarded as a burn-in period.

4.2 Posterior Results

Table 4 shows the posterior results of β, σ2t and k. The CD denotes the con-

vergence diagnostic statistic proposed by Geweke (1992).14 The sixth columnof Table 4 shows the p values for CD and the convergence of the posterior dis-tribution would be confirmed from them. Further, the posterior mean and the

11We obtained the data for the CPI from the web-site of the Bureau of Labor Statistics,U.S. Department of Labor (http://stats.bls.gov/).

12The posterior results obtained afterward are generated by using DIGITAL Visual Fortranversion 6.6 and all figures are drawn using Ox version 3.40 (Doornik 2001).

13These values are similar to those of Stephens (2000).14The CD can be defined as follows: for the given sequence {g(j) | j = 1, 2, · · · , ns}, if the

sequence is stationary,

CD =gA − gB

SA(0)/nA + SB(0)/nB

→ N (0, 1),

where

gA =1

nA

nAX

j=1

g(j), gB =1

nB

nsX

j=n∗g(j) (n∗ = ns − nB + 1),

and SA(0) and SB(0) denote consistent spectral density estimates. In this article, we set

nA = 3, 000 and nB = 15, 000, and calculate SA(0) and SB(0) using Parzen windows withbandwidths of 300 and 1, 500, respectively.

9

standard deviation of the number of components in the mixture k is about 23.7and 3.8, respectively. Using the following posterior interval

(posterior mean − 2 × posterior standard deviation,posterior mean + 2 × posterior standard deviation), (33)

roughly speaking, the number of components k varies from 16 to 31. Figure 2shows the estimated densities of x. It seems that the densities fit the histogramsof the data appropriately.

Table 5 shows the result of sensitivity analysis to choice of hyperparametervalues and to the number of distributions k in the mixture. In the table, thecase (a) has the hyperparameter values given in (32). The other cases changethe hyperparameter values as follows:

(b) κ = diag{1/R2t}, (c) κ = diag{0.01/R2

t}, (d) h = diag{200g/αR2t}

(e) h = diag{50g/αR2t}, (f) b = 0.2, (g) λ = 3.

The cases (b) to (e) give the similar value of k as in the case (a). Therefore,whether κ and h are more informative or less informative does not affect theposterior results. The case (f) of more informative value of b = 0.2 gives smallernumber of k than that in the case (a) (b = 0.002). The case (g) of λ = 3 giveslarger number of k than that in the case (a) (λ = 1). This result is the sameas a result in Stephens (2000, p.57). Figure 3 shows the estimated densities ofx in 1988 for the cases (b) to (g).15 From this figure, we can observe that theestimated densities are very similar, although the estimated density of case (f)is slightly steeper than that of the case (a).

Table 6 shows the posterior results for total, chronic and transient poverty.The higher the poverty line is, the higher the ratio of mean to standard deviationfor the poverty measures (mean/sd) is. When the poverty line is higher than3, 000, which is the minimal poverty line we set, all the ratios are higher than2. The range of poverty line in the table is not unrealistic, because the actualpoverty line in USA lies within the range we set, according to the web-site ofUS Census Bureau.16 This implies that the estimated poverty measures aresignificant and our method is appropriate for measuring poverty.

Figure 4 reports total, chronic and transient poverty for the data and pos-terior results. The horizontal and the vertical axes denote the poverty line andthe poverty measure. The solid lines, with the symbols ‘•’ and ‘+’, denote thepoverty measure computed by Ravallion’s method using the observed data andthe posterior mean of poverty measure, respectively. The dotted lines denotethe upper and the lower bounds of the posterior intervals (33). There are 5graphs for each ζ, which show the total and transient poverty derived from yand x, and the chronic poverty from x.

We compare the three types of poverty measure calculated by our methodwith those calculated by Ravallion’s method. It is observed, according to Figure4, that the transient poverty calculated by Ravallion’s method fluctuates and

15We have the same results of estimated densities in the other years as those in 1988.16The poverty threshold of a family that consists of two adults and one child, for exam-

ple, in 1988 is 9, 522 dollars according to the web-site. Using (31) and CPI, we obtain itsreal per capita value, deflated by equivalence scale, as approximately 4, 016 dollars. Seehttp://www.census.gov/.

10

at some points jumps out of the posterior interval based on both y and x, whenζ = 0.

FGT poverty measures with ζ = 0 or 1 are well-known poverty measures:the headcount ratio and the poverty gap. However, it is pointed out by Kurosaki(2005) that these measures have faults in that they do not reflect the degree ofpoverty well, and that the adoption of the FGT poverty measure with ζ > 1implies that the social planner is assumed to be risk-averse and inequality averse.Therefore, the FGT is often used as ζ = 2 (the squared poverty gap) in theliterature of development economics.17

Figure 5 shows the increment of the total and the chronic poverty measuresby Ravallion’s method defined as

ΔΠemF (Υl) = Πem

F (Υl) − ΠemF (Υl−1), l = 2, · · · , L (34)

ΔΠemC (Υl) = Πem

C (Υl) − ΠemC (Υl−1), l = 2, · · · , L, (35)

where ΠemF (Υl) and Πem

C (Υl) are defined in (23) and (24), and L is the numberof poverty lines. The total poverty and chronic poverty measures for ζ = 0fluctuate more than for ζ = 1 or 2 in the figure. It is interesting that thechronic poverty measure, which ought to be constant or move stably, fluctuatesaccording to the poverty line we assume when ζ = 0. These fluctuations causethe unstable movement of the transient poverty measures by Ravallion’s methodin Figure 4. Figures 4 and 5 suggest that the headcount ratio (the case of ζ = 0)may not capture poverty sufficiently, especially transient poverty.

5 Concluding Remarks

As shown in Ravallion’s seminal work, poverty can be decomposed into twocomponents, chronic poverty and transient poverty. This article provided analternative empirical method to measure them. Specifically, we proposed theBayesian mixture model with an unknown number of mixtures developed byStephens (2000). Our Bayesian approach in this article has the following merits:

1. No additional data (i.e., IV) are required for the estimation of EIV.

2. The Bayesian model in this article permits heterogeneity in individualincome by using the mixture model.

3. Unobserved incomes can be derived from the posterior results, and usingthem, we can use statistical inference on the total, chronic and transientpoverties.

In our real example using the PSID data, the ratios of posterior mean to pos-terior standard deviation are fairly high for the actual level of the poverty line.Furthermore, in most cases the posterior intervals include poverty measurescalculated using Ravallion’s descriptive statistics. Regarding that Ravallion’smethod has been supported, it shows that the plausible poverty measures canbe obtained by our method.

Acknowledgements The authors are grateful to Professor Khondaker Miza-nur Rahman of Nanzan University, Professor Fitzenberger, the editor, and three

17See, for example, Jalan and Ravallion (1998).

11

anonymous referees for their helpful comments, which improved an earlier ver-sion of this article. They also acknowledge the financial support of the JapaneseMinistry of Education, Culture, Sports, Science and Technology under Grant-in-Aid for Scientific Research No.13630024 and No.14653004, respectively.

Appendix

Stephens’ birth-death algorithm is as follows:18

Stephens’ birth-death algorithm Let define φj = (μj ,Σj) and ψ = {(w1, φ1),· · · , (wk, φk)} and the birth rate as λ.

1. Calculate the death rate for each component. The death rate for thejth component is given by

νj = λL(ψ \ (wj , φj))

L(ψ)p(k − 1)kp(k)

, j = 1, · · · , k,

where L(·) is the likelihood function.

2. Calculate the total death rate as ν =∑k

j=1 νj .3. Simulate the time to the next jump from an exponential distribution

with mean1

λ+ ν.

4. Simulate the type of jump: birth or death with respective probabili-ties

P(birth) =λ

λ+ ν, P(death) =

ν

λ+ ν.

5. Adjust ψ to reflect the birth or death:

birth : ψ ∪ (wb, φb)= {(w1(1 − wb), φ1), · · · , (wk(1 − wb), φk), (wb, φb)}

death : ψ \ (wj , φj) =

{(w1

1 − wj, φ1

), · · · ,

(wj−1

1 − wj, φj−1

),

(wj+1

1 − wj, φj+1

), · · · ,

(wk

1 − wj, φk

)}.

Birth: Simulate wb and φb independently from densities k(1−wb)k−1

and p(φb|θ), respectively.Death: Select a component to die: (wj , φj) ∈ ψ being selected withprobability

νj

νfor j = 1, · · · , k

6. Return to step 1.

In the birth-death process, we use a truncated Poisson prior with λ on thenumber of components k (Stephens, p.50, 2000):

p(k) ∝ λk

k!, k = 1, · · · , kmax = 100.

18Algorithm 3.1 in Stephens (2000).

12

References

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Diebolt J, Robert CP (1994) Estimation of finite mixture distributions throughBayesian sampling. Journal of the Royal Statistical Society B56:363–375

Doornik JA (2001) Ox 3.0: An object-oriented matrix programming language.Timberlake Consultants Ltd., London

Ferguson TS (1983) Bayesian density estimation by mixtures of normal distri-butions. In: Rizvi MH, Rustagi JS (eds.) Recent Advantages in Statistics:Papers in Honor of Herman Chernoff on His Sixtieth Birthday. AcademicPress, New York; 287–302

Florens JP, Mouchart M, Richard JF (1974) Bayesian inference in error-in-variables models. Journal of Multivariate Analysis 4:419–452

Foster J, Greer J, Thorbecke E (1984) A class of decomposable poverty mea-sures. Econometrica 52:761–766

Geweke J (1992) Evaluating the accuracy of sampling–based approaches to thecalculation of posterior moments. In: Bernardo JM, Berger JO, DawidAP, Smith AFM (ed.) Bayesian statistics 4. Oxford University Press,Oxford, 169–193

Gibson J (2001) Measuring chronic poverty without a panel. Journal of De-velopment Economics 65:243–266

Green PJ (1995) Reversible jump Markov chain Monte Carlo computation andBayesian model determination. Biometrika 82:711–732

Groves RM (1989) Survey errors and survey costs. Wiley, New York

Hasegawa H, Kozumi H (2001) Bayesian analysis on Engel curves estimationwith measurement errors and an instrumental variable. Journal of Busi-ness & Economic Statistics 19:292–298

Hasegawa H, Kozumi H (2003) Estimation of Lorenz curves: A Bayesian non-parametric approach. Journal of Econometrics 115:277–291

Hurn M, Justel A, Robert CP (2003) Estimating mixtures of regressions. Jour-nal of Computational and Graphical Statistics 12:55–79

Jalan J, Ravallion M (1998) Transient poverty in postreform rural China. Jour-nal of Comparative Economics 26:338–357

Kurosaki T, (2005) The Measurement of transient poverty: Theory and appli-cation to Pakistan. Institute of Economic Research, Hitotsubashi Univer-sity

Lewbel A (1996) Demand estimation with expenditure measurement errors onthe left and right hand side. The Review of Economics and Statistics78:718–725

13

Pittau MG (2005) Fitting regional income distributions in the European Union.Oxford Bulletin of Economics and Statistics 67:135–161

Pittau MG, Zelli R (2004) Testing for changing shapes of income distribution:Italian evidence in the 1990s from kernel density estimates. EmpiricalEconomics 29:415–430

Ravallion M (1988) Expected poverty under risk-induced welfare variability.The Economic Journal 98:1171–1182

Ravallion M, van de Walle D, Gautam M (1995) Testing a social safety net.Journal of Public Economics 57:175–199

Richardson S, Green PJ (1997) On Bayesian analysis of mixtures with anunknown number of components (with discussion). Journal of the RoyalStatistical Society B59:731–792

Stephens M (2000) Bayesian analysis of mixture models with an unknownnumber of components — An alternative to reversible jump methods. TheAnnals of Statistics 28:40–74

14

Table 1: Summary statistics

equivalence scale1988 1989 1990 1991 1992

mean 2.37395 2.38834 2.39820 2.39844 2.38234sda 0.31713 0.31249 0.30965 0.30923 0.30923m/sd 7.48563 7.64290 7.74484 7.75616 7.70405min 2.00426 2.00426 2.00426 2.00426 2.00426max 4.39901 4.39901 4.39901 4.13590 4.135902.50%b 2.00426 2.00426 2.00426 2.00426 2.004265% 2.00426 2.00426 2.00426 2.00426 2.0042625% 2.00426 2.00426 2.35524 2.35524 2.0042650% 2.35524 2.35524 2.35524 2.35524 2.3552475% 2.68503 2.68503 2.68503 2.68503 2.6850395% 2.99826 2.99826 2.99826 2.99826 2.9982697.50% 2.99826 2.99826 2.99826 2.99826 2.99826

Y1988 1989 1990 1991 1992

mean 16218.5 16383.5 16719.9 16802.8 17623.8sd 11531.0 12112.2 13057.1 13809.0 12673.8m/sd 1.40651 1.35264 1.28052 1.2168.0 1.39058min 704.8 700.4 190.9 280.6 613.3max 167643.3 190723.1 164896.3 266611.5 143240.42.50% 3286.4 2968.1 2777.7 2732.6 2920.05% 4307.1 4133.4 4008.3 4265.8 4391.925% 9152.1 9655.9 9518.2 9662.9 9981.250% 14060.9 14038.7 14287.8 13957.0 15010.375% 19733.5 19839.4 20569.8 20574.6 22330.895% 33961.1 35062.5 34970.6 34874.0 37624.797.5% 42243.2 43848.7 43327.7 43959.2 48542.3

y = log Y1988 1989 1990 1991 1992

sd 0.63807 0.64952 0.66858 0.67004 0.67972m/sd 14.88860 14.63348 14.23047 14.20222 14.07497min 6.55796 6.55166 5.25160 5.63680 6.41890max 12.02959 12.15858 12.01307 12.49355 11.872282.50% 8.09756 7.99568 7.92939 7.91300 7.979355% 8.36802 8.32686 8.29612 8.35838 8.3875225% 9.12174 9.17532 9.16097 9.17605 9.2084650% 9.55116 9.54957 9.56716 9.54374 9.6164975% 9.89007 9.89543 9.93158 9.93181 10.0137295% 10.43297 10.46489 10.46226 10.45950 10.5354297.50% 10.65120 10.68850 10.67655 10.69102 10.79019

a: ‘sd’ denotes the standard deviation.

b: ‘c%’ denotes the c percent point of data.

15

Table 2: Correlation matrix of y

1988 1989 1990 1991 19921988 1 0.87963 0.82729 0.7886 0.723561989 0.87963 1 0.86713 0.82506 0.753611990 0.82729 0.86713 1 0.86283 0.778491991 0.7886 0.82506 0.86283 1 0.823241992 0.72356 0.75361 0.77849 0.82324 1

Table 3: Total, chronic and transient poverty measures from data

ζ = 0Υ = eυ Total Chronic Transient3000 0.02532637 0.01436031 0.010966064000 0.04464752 0.03067885 0.013968675000 0.06657963 0.04699739 0.019582256000 0.09778068 0.07832898 0.01945177000 0.13250653 0.11031332 0.022193218000 0.17062663 0.15274151 0.017885129000 0.21710183 0.19843342 0.0186684110000 0.2689295 0.2421671 0.02676240



16

Table 4: Posterior results

meana sda mean/sd CDb CD p-valueβ11 1.72674 0.28653 6.02646 0.14767 0.88260β12 1.58823 0.2767 5.73981 -0.078920 0.93710β13 1.51405 0.26663 5.67847 -0.032986 0.97369β14 1.45526 0.26535 5.48425 0.14685 0.88325β15 1.36285 0.26095 5.22267 -0.012566 0.98997β22 1.71669 0.29081 5.90321 -0.48495 0.62771β23 1.58672 0.2783 5.70149 -0.19332 0.84670β24 1.52712 0.27597 5.53357 0.016996 0.98644β25 1.43597 0.2722 5.27534 -0.34426 0.73065β33 1.73896 0.29052 5.98561 -0.38570 0.69972β34 1.63252 0.28246 5.77963 -0.23942 0.81078β35 1.52599 0.27601 5.52869 -0.33643 0.73655β44 1.79868 0.29676 6.06108 -0.34411 0.73077β45 1.58585 0.28139 5.63577 -0.27149 0.78602β55 1.74932 0.2951 5.92786 -0.40052 0.68878σ2

1 0.0011 0.00067 1.63995 0.25149 0.80143σ2

2 0.00084 0.00044 1.88625 -1.8229 0.068325σ2

3 0.00089 0.00053 1.6649 -0.83710 0.40254σ2

4 0.00083 0.00043 1.94973 1.7216 0.085139σ2

5 0.00108 0.00072 1.4979 1.3147 0.18861k 23.7454 3.79746 6.25297 1.0764 0.28173

a: ‘mean’ and ‘sd’ denote the posterior mean and the posterior standard deviation,respectively.

b: ‘CD’ denotes the convergence diagnostic statistic of MCMC proposed by Geweke

(1992).

Table 5: Comparison of posterior results for k

meana sda mean/sd(a)b 23.7454 3.79746 6.25297(b) 23.1065 3.40147 6.79309(c) 23.9645 3.33524 7.18525(d) 22.9075 3.50345 6.53855(e) 23.2462 3.44160 6.75449(f) 19.4390 3.74489 5.19080(g) 29.1526 3.69131 7.89763

a: ‘mean’ and ‘sd’ denote the posterior mean and the posterior standard deviation,respectively.b: The case (a) has the hyperparameter values given in (32). The other cases changethe hyperparameter values as follows:

(b) κ = diag{1/R2t }, (c) κ = diag{0.01/R2

t }, (d) h = diag{200g/αR2t }

(e) h = diag{50g/αR2t }, (f) b = 0.2, (g) λ = 3.

17

Table 6: Total, chronic and transient poverty measures fromposterior results

ζ = 0

Υ = eυ Total (y) Total (x) Chronic Transient (y) Transient (x)

3000 meana 0.02731893 0.02727368 0.01590985 0.01140908 0.01136383sda 0.00317278 0.00317286 0.00291239 0.00122075 0.00121899

4000 mean 0.04681263 0.04672895 0.03166074 0.01515189 0.01506822sd 0.00421828 0.00421858 0.00407234 0.00141076 0.0014077

5000 mean 0.07173526 0.07159818 0.05333663 0.01839863 0.01826155sd 0.00523897 0.00524136 0.00519019 0.00156528 0.00156304

6000 mean 0.10265384 0.10245039 0.08156317 0.02109067 0.02088722sd 0.00624255 0.00624686 0.00629741 0.00167403 0.00166991

7000 mean 0.13973727 0.13947279 0.11671308 0.02302419 0.02275971sd 0.00723133 0.00724068 0.00741385 0.00174528 0.00173844

8000 mean 0.18254901 0.1822352 0.15853889 0.02401011 0.0236963sd 0.00815178 0.00816169 0.00849056 0.00183797 0.00183053

9000 mean 0.23008393 0.22974655 0.20614708 0.02393684 0.02359947sd 0.00895475 0.00896629 0.0094533 0.00191798 0.00191156

10000 mean 0.28094416 0.28061214 0.2580891 0.02285506 0.02252304sd 0.00959933 0.00961201 0.01022574 0.00199301 0.00198606

ζ = 1


3000 mean 0.00955136 0.0095358 0.00451094 0.00504042 0.00502486sd 0.00148649 0.00148617 0.00117329 0.00063994 0.0006394

4000 mean 0.0163228 0.01629547 0.00921085 0.00711195 0.00708462sd 0.0019771 0.00197681 0.00168675 0.00072034 0.00071965

5000 mean 0.02481727 0.02477371 0.01576494 0.00905233 0.00900877sd 0.00245645 0.00245632 0.00219325 0.00077997 0.00077913

6000 mean 0.03512734 0.03506278 0.02428456 0.01084278 0.01077822sd 0.00292255 0.0029226 0.00268945 0.00082228 0.00082146

7000 mean 0.04735069 0.04726171 0.03489508 0.01245561 0.01236662sd 0.00337605 0.00337656 0.00317369 0.00084773 0.00084691

8000 mean 0.06151954 0.06140511 0.04767046 0.01384907 0.01373465sd 0.00381591 0.00381692 0.0036488 0.00085892 0.00085838

9000 mean 0.07757048 0.07743194 0.06258645 0.01498403 0.01484549sd 0.00423826 0.00424002 0.00411166 0.00086173 0.00086155

10000 mean 0.0953435 0.09518496 0.0795098 0.0158337 0.01567516sd 0.00463689 0.00463943 0.0045558 0.0008606 0.00086084

a: ‘mean’ and ‘sd’ denote the posterior mean and the posterior standard deviation,

respectively.

18

Table 6: Continuedζ = 2


3000 mean 0.00496513 0.00495709 0.00197067 0.00299447 0.00298642sd 0.00096918 0.0009689 0.00067769 0.00049341 0.00049311

4000 mean 0.0084615 0.00844762 0.00409298 0.00436852 0.00435464sd 0.00128968 0.00128938 0.00099506 0.00055774 0.00055736

5000 mean 0.01282438 0.01280277 0.0071018 0.00572257 0.00570097sd 0.00160591 0.00160562 0.00131773 0.00060761 0.00060715

6000 mean 0.01806409 0.01803258 0.01103818 0.00702591 0.0069944sd 0.00191617 0.00191592 0.00163886 0.00064684 0.00064636

7000 mean 0.02421111 0.02416756 0.01594876 0.00826235 0.0082188sd 0.00221942 0.00221927 0.00195558 0.00067708 0.0006766

8000 mean 0.03129402 0.03123677 0.02187725 0.00941676 0.00935952sd 0.00251543 0.00251544 0.00226671 0.00069907 0.00069864

9000 mean 0.03932323 0.03925135 0.02884997 0.01047326 0.01040139sd 0.00280377 0.00280401 0.00257209 0.00071401 0.00071372

10000 mean 0.04828269 0.04819615 0.03686562 0.01141708 0.01133054sd 0.00308371 0.00308423 0.00287133 0.00072317 0.00072309

19

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1988

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1989

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1990

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1991

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1992

Figure 1: Histograms of y = log Y

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1988

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1989

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1990

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1991

6 7 8 9 10 11 12 13

0.25

0.50

0.75 1992

Figure 2: Estimated densities of x

20

6 7 8 9 10 11 12 13

0.25

0.50

0.75(a) (b)

6 7 8 9 10 11 12 13

0.25

0.50

0.75(a) (c)

6 7 8 9 10 11 12 13

0.25

0.50

0.75(a) (d)

6 7 8 9 10 11 12 13

0.25

0.50

0.75(a) (e)

6 7 8 9 10 11 12 13

0.25

0.50

0.75(a) (f)

6 7 8 9 10 11 12 13

0.25

0.50

0.75(a) (g)

Figure 3: Comparison of estimated densities of x a

a: The case (a) has the hyperparameter values given in (32). The other cases changethe hyperparameter values as follows:

(b) κ = diag{1/R2t }, (c) κ = diag{0.01/R2

t }, (d) h = diag{200g/αR2t }

(e) h = diag{50g/αR2t }, (f) b = 0.2, (g) λ = 3.

21

ζ = 0

3000 4000 5000 6000 7000 8000 9000 1000011000

0.1

0.2

0.3

Total (y)data posterior

3000 4000 5000 6000 7000 8000 9000 1000011000

0.1

0.2

0.3

Total (x)data posterior

3000 4000 5000 6000 7000 8000 9000 1000011000

0.1

0.2

0.3

Chronicdata posterior

3000 4000 5000 6000 7000 8000 9000 1000011000

0.01

0.02

Transient (y)data posterior

3000 4000 5000 6000 7000 8000 9000 1000011000

0.01

0.02

Transient (x)data posterior

ζ = 1

3000 4000 5000 6000 7000 8000 9000 1000011000

0.05

0.10


3000 4000 5000 6000 7000 8000 9000 1000011000

0.05

0.10


3000 4000 5000 6000 7000 8000 9000 1000011000

0.05

0.10 Chronicdata posterior

3000 4000 5000 6000 7000 8000 9000 1000011000

0.005

0.010

0.015

0.020 Transient (y)data posterior

3000 4000 5000 6000 7000 8000 9000 1000011000

0.005

0.010

0.015

0.020 Transient (x)data posterior

Figure 4: Total, chronic and transient povertya

a: The dotted lines denote the following posterior interval:

(posterior mean − 2 × posterior sd, posterior mean + 2 × posterior sd).

22

ζ = 2

3000 4000 5000 6000 7000 8000 9000 1000011000

0.025

0.050


3000 4000 5000 6000 7000 8000 9000 1000011000

0.025

0.050


3000 4000 5000 6000 7000 8000 9000 1000011000

0.02

0.04

Chronicdata posterior

3000 4000 5000 6000 7000 8000 9000 1000011000

0.005

0.010

Transient (y)data posterior

3000 4000 5000 6000 7000 8000 9000 1000011000

0.005

0.010

Transient (x)data posterior

Figure 4: Continued

3000 4000 5000 6000 7000 8000 9000 10000 11000

0.010

0.015

0.020

0.025

0.030

ζ=0Total Chronic

3000 4000 5000 6000 7000 8000 9000 10000 11000

0.002

0.004

0.006

0.008

ζ=1Total Chronic

3000 4000 5000 6000 7000 8000 9000 10000 11000

0.001

0.002

0.003

0.004

ζ=2Total Chronic

Figure 5: Fluctuation of total and chronic poverty on data byRavallion’s method

23

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